Modeling based on Petri-nets.
Lecture 8
1
High-level Petri nets
• The classical Petri net was invented by Carl Adam Petri in
1962.
• A lot of research has been conducted (>10.000 publications).
• Until 1985 it was mainly used by theoreticians.
• Since the 80-ties the practical use is increasing because of the
introduction of high-level Petri nets and the availability of
many tools.
• High-level Petri nets are Petri nets extended with
– color (for the modeling of attributes)
– time (for performance analysis)
– hierarchy (for the structuring of models, DFD's)
2
The classical Petri net model
A Petri net is a network composed of places ( ) and transitions ( ).
t2
t1
p2
p1
t3
p4
p3
Connections are directed and between a place and a transition.
Tokens ( ) are the dynamic objects.
The state of a Petri net is determined by the distribution of tokens
over the places.
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p1
p4
t1
p2
p3
Transition t1 has three input places (p1, p2 and p3) and two
output places (p3 and p4).
Place p3 is both an input and an output place of t1.
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Enabling condition
Transitions are the active components and places and tokens are passive.
A transition is enabled if each of the input places contains tokens.
t1
t2
Transition t1 is not enabled, transition t2 is enabled.
5
Firing
An enabled transition may fire.
Firing corresponds to consuming tokens from the input places and
producing tokens for the output places.
t2
t2
Firing is atomic.
6
Example
7
Non-determinism
t1
t2
Two transitions fight for the same token: conflict.
Even if there are two tokens, there is still a conflict.
8
Modeling
States of a process are modeled by tokens in places and state
transitions leading from one state to another are modeled by
transitions.
• Tokens represent objects (humans, goods, machines), information,
conditions or states of objects.
• Places represent buffers, channels, geographical locations,
conditions or states.
• Transitions represent events, transformations or transportations.
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Example: traffic light
red
yr
yellow
rg
gy
green
10
Two traffic lights
red1
red2
yr1
yr2
yellow1
rg1
gy1
yellow2
rg2
gy2
green1
green2
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Two safe traffic lights
red1
red2
safe
yr1
yr2
yellow1
rg1
gy1
yellow2
rg2
gy2
green1
green2
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Two safe and fair traffic lights
red1
red2
safe2
yr1
yr2
yellow1
rg1
yellow2
gy1
rg2
gy2
safe1
green1
green2
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Example: life-cycle of a person
child
puberty
marriage
bachelor
married
divorce
death
dead
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br
red
black
rr
bb
• The number of arcs between two objects specifies the number of
tokens to be produced/consumed.
• This can be used to model (dis)assembly processes.
15
Some definitions
• current state
The configuration of tokens over the places.
• reachable state
A state reachable form the current state by firing a sequence of enabled
transitions.
• dead state
A state where no transition is enabled.
br
red
black
rr
bb
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(3,2)
br
rr
red
black
bb\br
(1,3)
(3,1)
rr
br
bb\br
rr
bb
(1,2)
(3,0)
rr
bb\br
(1,1)
br
(1,0)
• 7 reachable states, 1 dead state.
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Exercise: your life-cycle
sleeping
start
stop
active
die
dead
• How many states are reachable?
• Is there a dead state?
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Exercise: readers and writers
begin
receive_mail
mail_box
rest
rest
type_mail
read_mail
send_mail
•
•
•
•
ready
How many states are reachable?
Are there any dead states?
How to model the situation with 2 writers and 3 readers?
How to model a "bounded mailbox" (buffer size =4)?
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High-level Petri nets
In practice the classical Petri net is not very useful:
• The Petri net becomes too large and too complex.
• It takes too much time to model a given situation.
• It is not possible to handle time and data.
Therefore, we use high-level Petri nets, i.e. Petri nets extended
with:
• color
• time
• hierarchy
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To explain the three extensions we use the following example
of a hairdresser's saloon.
hairdresser ready to begin
free
client waiting
start
waiting
finish
busy
ready
Note how easy it is to model the situation with multiple hairdressers.
21
The extension with color
A token often represents an object having all kinds of attributes.
Therefore, each token has a value (color) with refers to specific features of
the object modeled by the token.
name: Sally
age: 28
hairtype: BL
free
start
waiting
name: Harry
age: 28
experience: 2
finish
busy
ready
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Each transition has an (in)formal specification which specifies:
• the number of tokens to be produced,
• the values of these tokens,
• and (optionally) a precondition.
The complexity is divided over the network and the values of tokens.
This results in a compact, manageable and natural process description.
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Examples
c := a+b
a
b := -a
+
b
a
-
b
c
a >=0 | b :=  a
a
b
select
if a> 0
then b:= a
else c:=a
fi
a
sqrt
b
c
Exercise:
calculate |a+b| using these buiding blocks
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The extension with time
For performance analysis we need to model durations, delays, etc.
Therefore, each token has a timestamp and transitions determine the delay
of a produced token.
free
3
9
0
D=0
1
start
waiting
D=0
finish
D=3
busy
ready
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The extension with hierarchy
• A mechanism to structure complex Petri nets comparable to DFD's.
• A subnet is a net composed out of places, transitions and subnets.
h1
h2
waiting
ready
h3
free
start
busy
finish
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Exercise: remove hierarchy
h1
h2
waiting
ready
h3
free
begin
start
busy
pending end
finish
begin
pending end
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